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"In science there is only physics, rest are only stamp collecting"


"Mathematics is like a tree. It spreads with the branches and parts , which are dependent on other parts. And in this fashion it gets simple enough to wholly rest on a few principles. These are the roots of the tree"

Quantum mechanics

Schrodinger equation   | Matrix mechanics   |   Dirac Equation   |   Quantum electrodynamics    

Theory of relativity

Special theory of relativity   |   General theory of relativity

String theory


heigenberg picture of atom


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Matrix mechanics

This is most modern and precise theory of atomic phenomena. Matrix mechanics reveals that the occurances that happen inside atom are probalistic. We have to work with a collection of data instead of certain changes governed by cause and effect relationship. The position observable is a matrix which has an infinite number of elements each of which is a possible measureable quantity. Momentum observable is also a matrix. What we can best expect is that certain change is more likely than the other. Only statistical knowledge about the atom can help us determine which change is more likely than other. Thus a new kind of mathematics has been developed , which compels us to describe everything about the atom in terms of probability. This kind of uncertainty is inherent in the quantum reality. Heisenberg picture of atom is related to operators A and the Hamiltonian H which is itself an operator. The time evolution of such operator A is the basic equations of motion according to Heisenberg;


higenberg picture of atom

Eigenvector


Eigen vector

Eigenvector is a vector which produces the same vector multiplied by some constant when it is acted by some linear operator like a matrix A. In quantum mechanics the operator H (hamiltonian) acts on wave function eigenvector and produces a corresponding eigenvalue E.

Heisenberg's Uncertainty relation

The atomic model of Bohr had limitation due to its hypothetical orbits which can not be observed. Heisenberg thought this as a nonphysical description of the atom. So Heigenberg made a reasoning that the radiation that the electron emits can be taken as a physical observable. According to Bohr's theory an electron jumps from one orbit to another when it emits or absorbs light or electromagnetic energy without being in the intermediate space. The frequency of emitted radiation from an atom can be represented by two variable m and n , which represent two Bohr's orbits respectively. Comparing it with classical harmonic motion , which can be represented by Fourier series, Heisenberg showed that two observables x(t), p(t) can be correlated with this Fourier series. Each of these observables is a matrix (row m and column n). Thus atom can be completely modeled by two matrices : position and momentum matrices (with m rows and n columns) with frequency of radiation determined by plank's formula.


fourier frequency

The Fourier series representation of harmonic motion is :
fourier frequency

The equation of motion reasoned by Heigenberg has the following Fourier components which itself must oscillate at frequency W(mm) :
uncertainty principle

Observable quantities like phase, frequency and amplitude of radiation thus enter into the mathematical equation. But as the those are the quantities that depends on difference of two number, the position and momentum will be matrices of m rows and n columns. Sum of all the matrix elements thus has no physical significance as it represents just a wave.

Momentum and position matrix P(0), X(0)

At certain moment (t) the position and momentum has to take values as matrices:
uncertainty principle


uncertainty principle

As it is evident that two matrices does not always commute , there is no well defined position and momentum of any particle at the same instant. The order of product of two matrices changes the value of multiplication. If two matrices each have the same identical elements then the order of their product do not change the value of multiplication.
matrix multiplication


matrix multiplication

If each contains distinct elements , the order of multiplication changes the value of multiplication. As position and momentum, in general , do not contain identical elements , their product depend s on their order of multiplication. Such was the reasoning of Heisenberg. No state possesses definite position and momentum simultaneously. The next is general result of such anti-commutation. The result of the difference is number i (imaginary) multiplied by plank constant.


uncertainty principle

The product of variance of position and momentum , σ(x) and σ(p) can be proven to be less than h/2. This is the uncertainty that saves quantum mechanics from breaking down. A proof using standard deviation of operators can be made simply:

Any operator O that acts on wave wave function ψ must have standard deviation σ
uncertainty principle

uncertainty principle

uncertainty principle

uncertainty principle

uncertainty principle
More compact and popular version of the uncertainty principle is :
uncertainty principle
I

There is some misconception about the uncertainty principle. This is not related to the limitation of our observing instruments. Now matter how precise our measuring intruments are , there will be always some uncertainty or unpredictability. The uncertainty principle is intrinsic property of quantum world. This is the way our world in the smallest scale behaves. Due to this uncertainty principle particle and anti -particle always appear and annihilate in vacuum out of nothing. Classical objects like billiard balls, moon have definite position and momentum. We can determine their position and momentum simultaneouly. But when we try to measure the position of tiny objects like electron the very act of observation changes the momentum of the particle. Some momentum of the incident light is transferred to the electron. This change is random. As a result, we can not predict with certainty the exact momentum of the electron at the same time we measure its position.

Notes and additional comments

expectation value of a random experiment is the average value of all the outcomes of the experiment. Suppose you throw a six headed dice one time. Then the expectation value will be (1+2+3+4+5+6)/6 = 3.5. Similarly operators in quantum mechanics have expectation value. Operators act on observables to give outcome of quantum measurements. So it is natural that they have expectation values.

Quantum cosmology



matrix mechanics


Vector, scalar and tensor harmonics on three sphere are introduced in order to study gravitational physics. These harmonics are the eigenfunctions of covariant laplace operator which satisfy some divergence and trace identity ,and ortho-normality conditions.


tensor harmonics


The wave function is computed using these harmonics and corresponding Schrodinger equation can be formulated easily.

De sitter metric is a solution of Einstein's field equations. It was found by Astronomer De sitter. It is given as a static form which is like Swardchild's solution in case of non-rotating massive objects.


uncertainty principle


like swardchild solution de sitter metric contains a singularity at some r -value. The n-dimensional de Sitter space, denoted by dS(n), is the Lorenzian manifold which is the analog of n-sphere(with its Riemanian metric), and which has positive constant curvature and simply connected for at least n=3.


uncertainty principle



uncertainty principle


uncertainty principle


Quantum mechanics have been merged with special relativity successfully. This apparent marriage has brought out some difficulties. Qantum mechanics is regarded as a complete and most succesful theory. But this theory is not free of contradiction. It has resulted in a paradoxical phenomena called measurement paradox. Measuring a quantum system is depenent on the system measuring it. Observing a particular quantum phenomena does not imitate classical behavior. Prior to measurement the system posesses a myriad possibilities to be evident when measured. This is so called measurement paradox.



uncertainty principle



On the other hand when special relativity is combined with quantum mechanics , infinities creep into the calculution. It is not desirable to work with infinties in real world situation. In general theory of relativity singularities arise because of very dense mass and energy distribution. Laws of physics breaks down and can not predict anything in such a case.


wheeler ex-wife equation


wheeler Dr with equation is a functional differential equation to combine features of quantum mechanics with General relativity. A functional differential equation is a equation which relates a derivative of a functional to the change of the function on which the functional depends.


uncertainty principle



uncertainty principle


uncertainty principle



uncertainty principle



uncertainty principle



uncertainty principle



uncertainty principle



uncertainty principle


Einstein said that "God does not play dice" as he was an opponent of quantum mechanics. Stephen Hawking claimed that God not only play dice but He throws them where we can not see them. Here is a visualization of black hole


uncertainty principle


from where nothing can escape. Everything else which enters a black hole's event horizon becomes lost inside the singularity.
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Reference materials:


Fundamentals of quantum mechanics
A briefer history of time by S. Hawking
A brief history of time by S. Hawking
Quantum mechanics
Grand Design by Stephen Hawking
Higher Engineering Mathematics ( PDFDrive.com ).pdf
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